Categories: Mathematics, Physics.

Fundamental solution

Given a linear operator $$\hat{L}$$ acting on $$x \in [a, b]$$, its fundamental solution $$G(x, x')$$ is defined as the response of $$\hat{L}$$ to a Dirac delta function $$\delta(x - x')$$ for $$x \in ]a, b[$$:

\begin{aligned} \boxed{ \hat{L}\{ G(x, x') \} = A \delta(x - x') } \end{aligned}

Where $$A$$ is a constant, usually $$1$$. Fundamental solutions are often called Green’s functions, but are distinct from the (somewhat related) Green’s functions in many-body quantum theory.

Note that the definition of $$G(x, x')$$ generalizes that of the impulse response. And likewise, due to the superposition principle, once $$G$$ is known, $$\hat{L}$$’s response $$u(x)$$ to any forcing function $$f(x)$$ can easily be found as follows:

\begin{aligned} \hat{L} \{ u(x) \} = f(x) \quad \implies \quad \boxed{ u(x) = \frac{1}{A} \int_a^b f(x') \: G(x, x') \dd{x'} } \end{aligned}

While the impulse response is typically used for initial value problems, the fundamental solution $$G$$ is used for boundary value problems. Suppose those boundary conditions are homogeneous, i.e. $$u(x)$$ or one of its derivatives is zero at the boundaries. Then:

\begin{aligned} 0 &= u(a) = \frac{1}{A} \int_a^b f(x') \: G(a, x') \dd{x'} \qquad \implies \quad G(a, x') = 0 \\ 0 &= u_x(a) = \frac{1}{A} \int_a^b f(x') \: G_x(a, x') \dd{x'} \quad \implies \quad G_x(a, x') = 0 \end{aligned}

This holds for all $$x'$$, and analogously for the other boundary $$x = b$$. In other words, the boundary conditions are built into $$G$$.

What if the boundary conditions are inhomogeneous? No problem: thanks to the linearity of $$\hat{L}$$, those conditions can be given to the homogeneous solution $$u_h(x)$$, where $$\hat{L}\{ u_h(x) \} = 0$$, such that the inhomogeneous solution $$u_i(x) = u(x) - u_h(x)$$ has homogeneous boundaries again, so we can use $$G$$ as usual to find $$u_i(x)$$, and then just add $$u_h(x)$$.

If $$\hat{L}$$ is self-adjoint (see e.g. Sturm-Liouville theory), then the fundamental solution $$G(x, x')$$ has the following reciprocity boundary condition:

\begin{aligned} \boxed{ G(x, x') = G^*(x', x) } \end{aligned}

1. O. Bang, Applied mathematics for physicists: lecture notes, 2019, unpublished.

© Marcus R.A. Newman, a.k.a. "Prefetch". Available under CC BY-SA 4.0.